
96 MATHEMATICS
Notice that all the three
products of monomials, 3xy,
15xy, –15xy, are also
monomials.
TRY THESE
Can you think of two more such situations, where we may need to multiply algebraic
expressions?
[Hint: • Think of speed and time;
• Think of interest to be paid, the principal and the rate of simple interest; etc.]
In all the above examples, we had to carry out multiplication of two or more quantities. If
the quantities are given by algebraic expressions, we need to find their product. This
means that we should know how to obtain this product. Let us do this systematically. To
begin with we shall look at the multiplication of two monomials.
8.3 Multiplying a Monomial by a Monomial
Expression that contains only one term is called a monomial.
8.3.1 Multiplying two monomials
We begin with
4 × x = x + x + x + x = 4x as seen earlier.
Similarly, 4 × (3x) = 3x + 3x + 3x + 3x = 12x
Now, observe the following products.
(i) x × 3y = x × 3 × y = 3 × x × y = 3xy
(ii) 5
x × 3y = 5 × x × 3 ×
y = 5 × 3 ×
x × y = 15xy
(iii) 5x × (–3y) = 5 × x × (–3) × y
= 5 × (–3) × x × y = –15xy
Note that 5 × 4 = 20
i.e., coefficient of product = coefficient of
first monomial × coefficient of second
monomial;
and x × x
2
= x
3
i.e., algebraic factor of product
= algebraic factor of first monomial
× algebraic factor of second monomial.
Some more useful examples follow.
(iv) 5x × 4x
2
= (5 × 4) × (x × x
2
)
= 20 ×
x
3
= 20x
3
(v) 5x × (– 4xyz) = (5 × – 4) × (x × xyz)
= –20 × (x × x × yz) = –20x
2
yz
Observe how we collect the powers of different variables
in the algebraic parts of the two monomials. While doing
so, we use the rules of exponents and powers.
8.3.2 Multiplying three or more monomials
Observe the following examples.
(i) 2x × 5y × 7z = (2x × 5y) × 7z = 10xy × 7z = 70xyz
(ii) 4xy × 5x
2
y
2
× 6x
3
y
3
= (4xy × 5x
2
y
2
) × 6x
3
y
3
= 20x
3
y
3
× 6x
3
y
3
= 120x
3
y
3
× x
3
y
3
= 120 (x
3
× x
3
) × (y
3
× y
3
) = 120x
6
× y
6
= 120x
6
y
6
It is clear that we first multiply the first two monomials and then multiply the resulting
monomial by the third monomial. This method can be extended to the product of any
number of monomials.